Abstract

Multiple solutions of a nonlinear elliptic boundary value problem: a compuiter aided proof J. McKenna The purpose of this paper is to describe the first genuinely multidimensional result on a nonlinear elliptic boundary value problem which has been open since approximately 1981. We are concerned with the boundary value problem \begin{eqnarray} \begin{array}{rrl} \Delta u+u^2= s\phi_1(x) & ~~~~~in &~~ \Omega \\ u=0 & ~~~~~on &~~ \partial\Omega \end{array} \end{eqnarray} where $\phi_1(x)$ is the first eigenfunction of the Laplacian with Dirichlet boundary conditions. We will survey known results on this type of problem and show that on a square, this boundary value problem has at least four solutions for a large value of $s$. Approximate solutions are first found and then numerically validated. This paper describes joint work with Michael Plum of Karlruhe.

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